As is known, transmission reach in coherent optical transmission systems has been limited by both receiver sensitivity of modulation formats and fiber nonlinearities. Since the development/deployment of digital coherent receiver and digital transmitters enabled by high-speed analog-to-digital converters (ADC) and digital-to-analog converters (DAC) it has proven to be relatively straightforward to apply standard modulation formats—such as M-ary quadrature amplitude modulation (QAM) from well-developed wireless communications fields, to optical transmission thereby reaching different distances at corresponding spectral efficiencies. To further extend transmission reach, low-density parity check (LDPC) codes have been used to reduce the forward error correction (FEC) limit with soft-decision inputs to decoders. Unfortunately, the performance of the same LDPC codeword could perform differently in different modulation formats due to the bit mapping and constellation geometric shapes. In particular, for non-Gray-mapping constellation(s), it is desirable to design an optimized constellation to reduce the performance loss of LDPC.
In non-Gray mapping constellations, iterative decoding has been proposed in the art to take advantage of a tilted demodulator transfer curve in an extrinsic information transfer IEXIT) chart by continuing to exchange information between LDPC decoders and demappers. As the art has learned however, the complexity of this approach is too high to be implemented in any practical high-speed optical transceiver (See, e.g., J. X. Cai et al., “64 QAM based coded modulation transmission over transoceanic distance with >60 Tb/s capacity,” 2015 Optical Fiber Communications Conference and Exhibition (OFC), Los Angeles, Calif., 2015, pp. 1-3).
Accordingly, and instead of working directly on non-Gray mapping constellation(s), probabilistic-shaped constellation over Gray-mapping constellation (2m-QAM, m=even integers) have been proposed and demonstrated to achieve the Shannon limit at back-to-back (BTB) scenarios by allocating additional overhead to constellation shaping and FEC (See, e.g., Ghazisaeidi, Amirhossein, et al. “Advanced C+L-Band Transoceanic Transmission Systems Based on Probabilistically-Shaped PDM-64QAM”, Journal of Lightwave Technology (2017); Chandrasekhar, Sethumadhavan, et al., “High-spectral-efficiency transmission of PDM 256-QAM with parallel probabilistic shaping at record rate-reach trade-offs”, ECOC 2016-Post Deadline Paper; 42nd European Conference on Optical Communications, Proceedings of, VDE, 2016). Due to the impact of fiber nonlinearity however, the probabilistic-shaped QAM suffers more nonlinearity penalties, thus making its transmission reach not as expected in the BTB case. Moreover, the encoding and decoding of non-uniform symbols are not trivial to be implemented in real-time high-speed coherent optical transceivers.
Geometrical shaping has been designed for Gray-mapping—such as amplitude-phase shift keying (APSK) for linear back-to-back noise case(s). Even though GMI capacity of geometrical shaping is better than standard QAM (See, e.g., S. Zhang et al., “Capacity-Approaching Transmission Over 6375 km Using Hybrid Quasi-Single-Mode Fiber Spans”, in Journal of Lightwave Technology, vol. 35, no. 3, pp. 481-487, Feb. 1, 2017), the nonlinearity tolerance after transmission is not guaranteed because the fiber nonlinearity is modulation dependent.